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Analytical Solutions to the Stochastic Kinetic Equation for Liquid and Ice Particle Size

Spectra. Part I: Small-size fraction

Vitaly I. Khvorostyanov

Central Aerological Observatory, Dolgoprudny, Moscow Region, Russian Federation

Judith A. Curry

School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia

(Manuscript received XX Month 2007, in final form XX Month 2007)

_________________________________________

Corresponding author address:

Dr. J. A. CurrySchool of Earth and Atmospheric SciencesGeorgia Institute of TechnologyPhone: (404) 894-3948Fax: (404) 894-5638curryja@eas.gatech.edu

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Abstract

The kinetic equation of stochastic condensation for cloud drop size spectra is extended to account

for crystalline clouds and also to include the accretion/aggregation process. The size spectra are

separated into small and large size fractions that correspond to cloud drops (ice) and rain (snow).

In Part I, we derive analytical solutions for the small-size fraction of the spectra that correspond

to cloud drops and cloud ice particles that can be identified with cloud liquid water or cloud ice

water content used in bulk microphysical schemes employed in cloud and climate models.

Solutions for the small-size fraction have the form of generalized gamma distributions. Simple

analytical expressions are found for parameters of the gamma distributions that are functions of

quantities that are available in cloud and climate models: liquid or ice water content and its

vertical gradient, mean particle radius or concentration, and supersaturation or vertical velocities.

Equations for the gamma distribution parameters provide an explanation of the dependence of

observed spectra on atmospheric dynamics, cloud temperature, and cloud liquid water or ice

water content. The results are illustrated with example calculations for a crystalline cloud. The

analytical solutions and expressions for the parameters presented here can be used for

parameterization of the small-size fraction size spectra and related quantities (e.g., optical

properties, lidar and radar reflectivities).

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1. Introduction

Parameterization of cloud microphysics in a bulk cloud model or general circulation

model (GCM) is usually based on the use of some a priori prescribed analytical functions that

describe the shape of the drop or crystal size spectra. Such functions are typically selected from

known empirical fits to the observed size spectra. Thorough reviews of these parameterizations

and the experimental data on which they are based are given by Cotton and Anthes (1989),

Young (1993), and Pruppacher and Klett (1997, hereafter PK97).

Generalized gamma distributions have been found to provide a reasonable approximation

for cloud and fog drops with modal radii of a few microns (e.g., Levin 1954; Dyachenko 1959)

)exp()( λβrrcrf pN −= . (1.1)

Here, p > 0 is the spectral index or the shape parameter of the spectrum, β and λ determine its

exponential tail, and cN is a normalization constant. It is often assumed that λ = 1,

)exp()( rrcrf pN β−= , (1.2)

which is referred to as a simple gamma distribution.

The index p determines the spectral dispersion of the gamma distribution (1.2)

( )2/1

02 )(1

−= ∫

∞drrfrr

rNrσ 2/1)1( −+= p , (1.3)

Typical values for liquid clouds of p = 6 - 12 were measured by Levin (1954) and confirmed in

many subsequent experiments (e.g. PK97). These values of p correspond to values of the relative

spectral dispersion σr = 0.38 - 0.28.

It was found that the inverse power law,

pN rcrf =)( , 0<p , (1.4)

can approximate the spectrum of larger drops in some liquid clouds in the radius range from 100

to between 300 and 800 µm with p = -2 to - 12 (Okita 1961; Borovikov et al. 1965; Nevzorov

1967; Ludwig and Robinson 1970). The expression (1.4) has also been used to represent ice

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crystal size spectra in cirrus and frontal clouds in the intermediate size region from ~20 to

between 100 and 800 µm, with values of p in the range -2 to -5 (e.g. Heymsfield and Platt 1984,

hereafter HP84; Platt 1997; Ryan 2000). Note that (1.4) is a particular case of the gamma

distribution (1.2) with β = 0.

These empirical spectra are widely used in the remote sensing of clouds (e.g., Ackerman

and Stephens 1987; Matrosov et al., 1994; Sassen and Liao 1996; Platt 1997) and also in the

modeling and parameterization of cloud properties and processes (e.g., Kessler 1969; Clark 1974;

Lin et al. 1983; Rutledge and Hobbs 1983; Starr and Cox 1985; Cotton et al. 1986; Mitchell 1994;

Pinto and Curry 1995; Fowler et al. 1996; Ryan 2000). A new impetus for bulk microphysical

models was given recently by development of the double-moment bulk schemes that include

prognostic equations for the number concentrations of hydrometeors in addition to the mixing

ratio, which allows a higher accuracy in predicting the cloud microphysical properties (e.g.,

Ferrier 1994; Harrington et al. 1995; Meyers et al. 1997; Cohard and Pinty 2000; Khairoutdinov

and Kogan 2000; Girard and Curry 2001; Morrison et al. 2005a,b, Milbrandt and Yau 2005a, b).

A deficiency of empirically-derived parameterizations of particle size spectra is that the

parameters are generally unknown and are fixed to some prescribed values. Experimental studies

show wide variations of the spectral dispersions (i.e., indices p) in clouds (e.g., Austin et al. 1985;

Curry 1986; Mitchell 1994; Brenguier and Chaumat, 2001). Morrison et al. (2005a,b) and

Morrison and Pinto (2005) calculated the values of p in a double-moment bulk microphysics

scheme using analytical expressions from Khvorostyanov and Curry (1999b); these values

showed substantial variations in time and space, resulting in variations in the onset of coalescence

and in cloud optical properties.

Several attempts have been made to derive these spectra from kinetic equations or from

equivalent approaches. Buikov (1961, 1963) formulated the first kinetic equations for the

condensation growth and obtained their time-dependent analytical solutions in the diffusion and

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kinetic regimes. Srivastava (1969) derived the generalized gamma distribution (1.1) with p = 2

and λ = 3 for drops under the assumption that the cloud space is divided into a number of cells

and the mass growth rate of each drop by diffusion is proportional to the volume of each cell.

Various versions of the stochastic kinetic equations of condensation have been derived

(Levin and Sedunov 1966, Sedunov 1974, Voloshchuk and Sedunov 1977, Manton 1979, Jeffery

et al. 2007; see reviews in Cotton and Anthes 1989; Khvorostyanov and Curry 1999a,b; Jeffery et

al. 2007) and the other approaches have been developed (e.g. Liu, 1995; Liu et al., 1995; Liu and

Hallett, 1998; McGraw and Liu, 2006), which provide solutions for the small drop fraction in the

form of gamma distribution (1.1) with fixed values of p and λ.

While these papers were able to derive gamma distributions similar to the observed

droplet size spectra, they had the following deficiencies: the spectral parameter p was always

fixed and the corresponding dispersions were constant and too high; the power of r in the

exponential tail in (1.1) was always λ = 2 or 3, which is larger than the often observed value of λ

= 1 (PK97); and none of these solutions was applied to crystalline clouds.

Khvorostyanov and Curry (1999a,b, hereafter KC99a, KC99b), derived a version of the

kinetic equation of stochastic condensation that accounted for the nonconservativeness of

supersaturation fluctuations, which has a solution in the form of simple gamma distribution for

particles with negligible fall velocities. In this paper, the kinetic equation from KC99a is extended

to treat conditions relevant to ice clouds and account for the coalescence/aggregation process,

allowing simultaneous consideration of condensation/deposition and aggregation. Analytical

solutions are derived to the kinetic equation under various assumptions.

Part I addresses the small-size fraction and the companion paper by Khvorostyanov and

Curry (2007, hereafter Part II) considers the large-size fraction. In particular, it is shown that all

the aforementioned empirical distributions (generalized gamma, exponential and inverse power

law) and their mixtures can be obtained as analytical solutions to the kinetic equation under

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various circumstances that include condensation and deposition, sedimentation, and aggregation.

These results provide an internally consistent description of cloud particle spectra in both liquid

and crystalline clouds over the extended size range of particles, providing a framework for

relatively simple parameterization schemes that allows for variation in spectral dispersion

associated with the bulk environmental characteristics. Part I is organized as follows. Section 2

presents the basic equations and assumptions. In sections 3 and 4, the analytical solutions to the

kinetic equations are obtained for various cases. Physical interpretation of the solutions is given

in section 5, and an example of calculations for a crystalline cloud is presented in section 6.

Section 7 is devoted to the summary and conclusions.

2. Basic equations, assumptions and simplifications

The basic equations developed here are applicable to pure liquid or pure crystalline

clouds; mixed phase clouds are not explicitly addressed here. The kinetic equation of stochastic

condensation derived in KC99a can be generalized by adding the collection (aggregation) and

breakup terms (∂f/∂t)col and (∂f/∂t)br as

tf

∂∂ ( )[ ]frvu

x iii

3)( δ∂∂

−+ ( )frr cond&

∂∂

+

fr

HGx

kr

HGx Rj

jijLi

i

+

+= 33 ∂

∂δ

∂∂

∂∂

δ∂∂ sr

coltf

+

∂∂

brtf

+

∂∂ , (2.1)

where it is assumed that both liquid drops and ice crystals are represented by spheres of radius r.

Here summation from 1 to 3 is assumed over doubled indices, t is time, xi and ui are the 3

coordinates and components of air velocity, v(r) is the particle fall velocity, condr& is the

condensation (deposition) growth rate, δi3 is the Kroneker symbol, and G is a parameter defined

following KC99a. We introduce here the left operator LHr

, and the right operator RHs

, such that

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nijijL kkH =

r, Hkk ij

nij

s= , nn

ijRijL kHkH =sr

. These operators convert the conservative tensor, kij,

into nonconservative tensors, kijn, kij

nn, and commute with operators ∂/∂xi and ∂/∂r.

The conservative tensor kij is defined here as in the statistical theory of turbulence and is

related to the velocity correlation function Bij(t) that is expressed as the integral expansion over

the turbulent frequencies ω (e.g., Monin and Yaglom 2007)

ωωω dFetB ijti

ij )()( ∫∞

∞−

= ∫∞

=0

)()(cos ωωω dEt ij , (2.2)

dttBkLT

ijij ∫=0

)( (2.3)

where Fij(ω) is the spectral function and Eij(ω) = 2Fij(ω) is the spectral turbulent density defined

as in Monin and Yaglom (2007), TL is the Lagrangian correlation time. The nonconservative

tensors kijn, kij

nn were derived in KC99a by generalizing the Prandtl’s mixing length conception

for the 4D space ),( rxr and performing Reynolds averaging of the kinetic equation over the

ensemble of turbulent fluctuations of the spectrum f′, supersaturation S′, and growth rate condr'&

and evaluating the correlations >< ''Sf , >< '' fr cond& , >< '' Sr cond& , >< '' SS with account for

nonconservativeness of supersaturation in fluctuations. Then kijn arise as the time integral of the

corresponding nonconservative correlation function Bijn that describes the correlation Su j ′′

),( pnij tB τ =

−−∞

∞

∫ ei i

F di t p

pij

ω ω

ω ωω ω

( )( )

∫∞

+

+=

02 )(

)/(1

)]sin()/()[cos(ωω

ωω

ωωωωdE

ttij

p

p , (2.4a)

∫=LT

pnijp

nij dttBk

0

),()( ττ . (2.4b)

Here τp is the supersaturation relaxation time,

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1)4( −= rNDvp πτ , (2.5)

Dv is the vapor diffusion coefficient, N is the number concentration, r is the mean radius, and we

introduced the inverse quantity rNDvpp πτω 41=

−= that can be called the “supersaturation

relaxation frequency”. The functions Bijnn and kij

nn arise from the autocorrelation ′ ′S t S t( ) ( )1 of a

nonconservative substance S′:

∫∞

∞− += ωω

ωω

ωτ ω dFetB ij

p

ptip

nnij )(

)(),( 22

2

∫∞

+=

02 )(

)/(1)cos( ωω

ωωω dEt

ijp

, (2.6)

∫=LT

pnnijp

nnij dttBk

0

),()( ττ . (2.7)

In (2.1), the correlation functions and turbulent coefficients were derived in KC99a for

pure liquid clouds; however, we have verified that the equations for pure crystalline clouds have

the same form as the expressions derived previously for pure liquid clouds. The major difference

between pure liquid and crystalline clouds is in the different values of supersaturation S, (over

water or ice in liquid or crystalline clouds), and in relaxation times τp caused by the different N,

r . The times τp determine the rate of supersaturation absorption and the “degree of

nonconservativeness” of the substance interacting with supersaturation (the drops or crystals),

i.e., the magnitude of kijn(τp), kij

nn(τp). The values of τp ~ 1 - 10 s for liquid clouds with N ~ 1-

5×102 cm-3 and r ~ 5 - 10 µm. Calculations of τp in cirrus and mid-level ice clouds using spectral

bin models show that it varies in the range 102 - 104 s for typical N and r (e.g., Khvorostyanov

and Sassen 1998, 2002; Khvorostyanov, Curry et al. 2001). Thus, the values of kijn(τp), kij

nn(τp)

are smaller for crystalline clouds than for liquid clouds but estimation from the above equations

shows that values of kijn(τp), kij

nn(τp) are comparable for both liquid and crystalline clouds.

Analytical solutions to the kinetic equation (2.1) can be obtained with the following

additional assumptions and simplifications:

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1) The cloud is quasi-steady state ∂f/∂t = 0. This does not imply complete steady state,

and the time dependence can be accounted for via the integral parameters in the solution.

2) The cloud is horizontally homogeneous, ∂f/∂x = ∂f/∂y = 0.

3) The size spectrum is divided into 2 size regions,

><

=,),(,),(

)(0

0

rrrfrrrf

rfl

s (2.8)

where fs(r) and fl(r) are the small-size and large-size fractions, and r0 is the boundary radius

between the two size fractions. The value of r0 ~ 40 - 60 µm for the drops and ~ 100 - 400 µm for

the crystals separates the domain where accretion rates are small from the domain where terminal

velocities are sufficiently large that collection becomes important (see e.g., Cotton and Anthes

1989, pp. 90-94). The functions fs(r) and fl(r) corresponds to the bulk categorization of the

condensed phase into cloud water and rain for the liquid phase and cloud ice and snow for the

crystalline phase.

4) The nonconservative turbulence coefficients are parameterized following KC99a to be

proportional to the conservative coefficient k as kijn = cnk, and kij

nn = cnnk. In general, cn, cnn have

values that are close to unity.

5) The coagulation term (∂f/∂t)col is described in detail in Part II and we assume that drop

breakup does not influence the small fraction. The coagulation-accretion growth rate (both for the

drop collision-coalescence and for the crystal aggregation with crystals) is considered in the

continuous growth approximation (Cotton and Anthes 1989; PK97, Seinfeld and Pandis 1998),

with account for only the collision-coalescence between the particles of the different fractions of

the spectrum, fs(r) and fl(r). The decrease in small fraction fs is described in this approximation as

collection of the small-size fraction by the large-size fraction.

scollosscol

s fItf

σ−=−=

∂∂ , 0rr < , (2.9)

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∫∞

=0

)()(2

rlccol drrfrvrEπσ (2.10)

These equations are derived in Part 2 from the Smoluchowski stochastic coagulation equation

along with the mass conservation at the transition between the small- and large-size fractions.

With these assumptions, the kinetic equation (2.1) can be written for fs:

( )[ ]sfrvwz

)(−∂∂ ( )scond fr

r&

∂∂

+ 2

2

zfk s

∂∂

=

+

∂+

rzf

zrfkGc ss

n ∂∂∂

∂∂ 22

2

22

rfkGc s

nn∂

∂+ scol fσ− , (2.11)

All of the quantities on the left-hand side (f, w, S) indicate averaged values in the sense discussed

in KC99a,b, and all fluctuations are included on the right-hand side of the equation. The terms on

the left-hand side describe convective and sedimentation fluxes and drop (crystal) condensation

(depositional) growth/evaporation, and the terms on the right-hand side describe turbulent

transport, stochastic effects of condensation/deposition growth and absorption by the large-size

fraction.

To enable analytic solutions to (2.11), we make the following additional assumptions:

6) The vertical gradient of fs can be parameterized as a separable function:

),()()(),( zrfrzz

zrfss

s ςα=∂

∂ . (2.12)

The functions α s(z) and the ζs(r) can be different for each fraction, determining the magnitude of

the gradient and its possible radius dependence. If ζs(r) = 1, and dfs/dz = αsf, it is easily shown

from the definition of liquid (ice) water content that

)/)(/1()( dzdqqz lslss =α . (2.13)

7) We assume that terminal velocity v(r) can be parameterized following Khvorostyanov

and Curry (2002, 2005) in the form

)()()( rBv

vrrArv = , (2.14)

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where Av(r) and Bv(r) are continuous functions of particle size, that include consideration of

various crystal habits and non-spherical drops and turbulent corrections to the flow around the

particles.

8) The condensation or deposition growth/evaporation rate condr& is described by the

simplified Maxwell equation

rbSrcond =& ,

wp

vsvDbρρ

Γ= , (2.15)

where S = (ρv - ρvs)/ρvs is supersaturation over water (ice), ρv and ρvs are the environmental and

saturated over water (ice) vapor densities, ρw is the water (ice) density, and Γp is the

psychrometric correction associated with the latent heat of condensation.

Substituting (2.12) - (2.15) into (2.11) allows elimination of z- derivatives and we have

a differential equation that is only a function of r:

2

22

drfdkGc s

nn drdf

rbSrkGc s

ssn

−+ )(2 ςα

0)()(2 22 =

+−−−−+++ scols

ss

ssn f

rbS

dzdwvw

dzkdk

drdkGc σα

αα

ςα . (2.16)

The dimensionless parameter G was derived in KC99a,b in the process of averaging over the

turbulence spectrum, and can be written in various forms:

)4( rNDLc

rDG

v

wdp

w

avπ

γγρρ −

= 24 rNLc wdp

w

a

πγγ

ρρ −

=

ls

wdpa

qr

Lc )( γγρ −

≈adlsls dzdqq

r)//(

=1

,

−

⋅=

adls

lsq

qzr , (2.17)

where cp is the specific heat capacity, L is the latent heat of condensation or deposition, γd and γw

are the dry and wet (water or ice) adiabatic lapse rates, ρa is the air density, N and r are number

concentration and mean radius of the particles, ql is the liquid (ice) water content and we have

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introduced the vertical gradient of the adiabatic liquid (ice) water content (e.g. Curry and Webster

1999)

( ) awdp

ad

lsL

cdz

dqργγ −=

(2.18)

The quasi-steady supersaturation Sq can be written following KC99a,b :

efpq wAS τ= , )( wdvs

ap

p

Lc

A γγρρ

−Γ= , (2.19)

where τp is the supersaturation relaxation time (2.5) and the wef is the effective vertical velocity

described by KC99a,b. Then the condensation term (2.15) can be rewritten as

rwc

rbS

r efconqcond ==& , (2.20)

rNDLc

DbArGcv

wdp

w

avpcon π

γγρρ

τ4

)( −=== . (2.21)

3. Solution neglecting the diffusional growth of larger particles

In some situations, it can be assumed that the terms with diffusional growth can be

neglected for the larger particles in the small-size fraction. These terms can be much smaller in

the tails of the spectra than the other terms in (2.16) for small values of super- or subsaturation, or

for sufficiently large gradients αs in relatively thin clouds, or for sufficiently large values of G.

This allows for neglect in (2.16) of the diffusional growth terms in the tails of the spectra since

they decrease with radius faster than the other terms. We obtain asymptotic solutions in the small

and large particle limits, and then merge these two solutions.

3.1 Solution at small r

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The analysis of asymptotic behavior of the individual terms in (2.16) for small values of

r shows that the most singular in r are the 1st, 3rd and last terms, and hence we retain in (2.16)

only these terms:

2s

22

drfdr

drdfra s

1+ 01 =− sfa , (3.1)

21 kGcbSa

nn−= . (3.2)

Eq. (3.1) is a linear homogeneous Euler equation of 2nd order and its solution can be obtained in

the form of a power law (Landau and Lifshitz 1958):

ps rrf ~)( . (3.3)

Substitution of (3.3) into (3.1) yields

2kGcbSp

nn= . (3.4)

In cloud growth layers with positive supersaturation, S > 0, then p > 0, and (3.3) has the

form of the left branch of the gamma distribution (for r smaller than the modal radius). In

evaporating clouds with negative supersaturation, S < 0, then p < 0 and the solutions have the

form of the inverse power laws. Eq. (3.4) is a generalization of the corresponding solution from

KC99b where the index p was expressed via the mean effective vertical velocity under

assumption of quasi-state state supersaturation. Now, (3.4) accounts for various possible sources

of supersaturation (advective, convective, radiative, mixing among the parcels and with

environment).

3.2 Solution at large r

Here we consider the larger particles in the small-size fraction, where r < r0, with r0

being the boundary between the small- and large-size fractions defined in section 2 in (2.8). We

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assume that supersaturation and diffusional growth are sufficiently small that we can eliminate in

(2.16) the condensation growth terms that are proportional to r-1 and ~r-2. Then (2.16) becomes

2

22

drfdkGc s

nn drdfkGc s

sn α2+

[ ] 0)1(2 22 =−++ ssssn fkkGc µαα , (3.5)

where

2/1

2 )(1

−++−=

dzkd

dzdwvw

ks

colss

sα

σαα

µ . (3.5a)

We seek a solution similar to KC99b as the exponential tail of the gamma distribution:

)exp()( rrf ss β−∝ . (3.6)

Substitution into (3.5) yields a quadratic equation for βs

0)1(2 2222 =−+− ssssnsnn kkGckGc µαβαβ , (3.7)

which has two solutions for βs

( )kGc

kckckc

nn

ssnnsnsns

2/1222222

2,1,)1( µααα

β−−

=m . (3.8)

The solutions are simplified if the nonconservativeness of k-components is neglected, cnn = cn = 1:

)1(2,1, ss

s Gµ

αβ m= . (3.9)

Hereafter in this section, the negative and positive signs relate to the 1st and 2nd solutions

respectively. Eq. (3.6) represents the exponential tail of the spectrum and the slopes (3.8), (3.9)

are generalizations of the corresponding expression from KC99b. Physical conditions require that

βs,1,2 > 0.

3.3 Merged solution

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From the two asymptotic solutions, the general solution to (2.16) can be found following

KC99b to be of the form

)()exp()( 2,1,2,1 rrrcrf sp

s Φ−= β . (3.10)

Substituting (3.10) into (2.16) yields the confluent hypergeometric equation for Φ with two

solutions that are Kummer functions (described in detail in Appendix A). Then the complete

solutions for these cases with correct asymptotics at small and large r are

−= )1(exp)( 2,12,1, s

sps G

rcrf µα

m

±× r

GppF ss

ss

µαµ

µ2;),1(

2m , (3.11)

where F(a,b;x) is the Kummer (confluent hypergeometric) function. These solutions are derived

in Appendix A assuming v = const and µs = const. Now, the correction for v(r) can be introduced

into (3.11) using the real v(r), and the solution is valid at such r that µs2 > 0. Evaluation of the

normalizing constants c1,2, moments and some properties of these solutions are described in

Appendix A.

Equations (3.11) are similar to gamma distributions with some modifications that

generalize the solutions from KC99b. The left branch of these gamma distribution type spectra is

described by the index p, which is now sign-variable, depending on supersaturation and allows

for various sources of cooling. It was found in KC99b that there are 2 asymptotics of solutions of

the type (3.11) for sufficiently small or large qls (e.g., near the boundaries or the center of the

cloud), where the Kummer functions are reduced to the product of the power law and exponent

and (3.11) is reduced to the gamma distribution (3.10) with Φ = 1 but with modified p, βs. These

asymptotics can be generalized using the new formulation to account for sedimentation, accretion

and gradients of w and k.

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4. Solution including the diffusional growth of large particles

The solution obtained in section 3 assumed that the diffusion growth terms are small at

sufficiently large r. However, if these terms are still significant in the tail of the small-size

fraction, as it can be at large sub- and supersaturations, we must seek another solution that

accounts for these terms. We again consider the two asymptotic cases at small and large r (again,

within the small-size fraction).

At small r, the same most singular terms in (2.16) give the main contribution, and the

solution is the same power law (3.3), fs ~ rp, with the index p defined in (3.4). At large r, if the

diffusional growth terms are retained, then (2.16) is a 2nd order differential equation with rather

complicated variable coefficients, and an analytical solution is not easily obtained. Hence, we

seek a simplification that will enable an analytical solution. At large r, it is reasonable to assume

that the tail of the spectrum is smooth and the r-gradients of the spectrum are smaller than near

the mode. Then we can neglect in (2.16) the stochastic diffusion terms in radii-space (the terms

with G), and arrive at the following equation

sss fdzdwfrvw +− α)]([

+

rf

drdbS s

scols

ss fdz

dkdzdkk

−++= σ

ααα 2 (4.1)

Equation (4.1) can be rewritten as

ss fr

rf

drd )]([ 21 ξξ +=

, (4.2)

where

−−−++= cols

sss dz

dwwdz

dkdzdkk

bSσα

αααξ 2

11 , (4.3a)

bSrvs /)(2 αξ = . (4.3b)

Introducing a new variable ϕs = fs(r)/r, (4.2) can be rewritten as

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ss rr

drd

ϕξξϕ )]([ 21 += . (4.4)

Integration from some r1 to r yields

)exp()()( 111 ssss JJrr += ϕϕ , (4.5)

where

∫=r

rs rdrJ

1

11 ξ2

)( 12

12 ξrr −

= , (4.6)

∫=r

rs rdrrJ

1

)(22 ξ2

)()( 2112

22

+−

=vB

rrrr ξξ . (4.7)

When evaluating Js2, we assume that v(r) at this size range can be approximated by the power law

(2.14) with constant Av, Bv. Substituting these integrals into (4.5), and again using fs = ϕs/r, we

obtain the solution for the larger portion of the small fraction r < r0:

})()([exp{)()( 2112

221

1rrrrrf

rrrf ssss ββ −−= . (4.8)

where the slope βs2 is

+

+−=22

212

vs B

ξξβ

+−

−−−++=

2)(

2''/1 2

v

sssscolsB

rvkkkdzdwwbS

αααασα , (4.9)

and the primes denote here derivatives by z. This is the solution for the tail of the small fraction

expressed via its value at r = r1.

Eq. (4.8) can be written in a slightly different form that includes the terms with r1 in the

normalizing constant cN

])(exp[)( 22 rrrcrf sNs β−= . (4.10)

The merged solution can be constructed again as in section 3 as an interpolation between the two

asymptotic regimes at small and large r:

18

)(])(exp[)( 22 rrrrcrf s

pNs Φ−= β . (4.11)

Unfortunately, the resulting equation for Φ in this case is much more complicated than the

confluent hypergeometric equation in section 3 and its solutions cannot be reduced to Kummer

functions. Thus, (4.11) can be used with some interpolation formulae for Φ, e.g.,

)/tanh()/()/exp()( 1sc

pscsc rrrrrrr −+−=Φ , (4.12)

where rsc is a scaling radius comparable with r . At r << rsc, the first term tends to 1, the 2nd term

tends to 0, and we get obtain from (4.11) fs ~ rp at small r since βs2(r)r2 << 1 in (4.11) and exp →

1. At r >> rsc, the solution tends to (4.10). Thus, (4.12) ensures correct limits at both small and

large r.

An advantage of (4.11) is that the tail of the spectrum explicitly accounts for the

diffusional growth process. This results in the inverse dependence of the slope βs2 (4.9) on

supersaturation,; that is, as |S| increases, the slope becomes steeper. This is physically justified

since more vigorous condensation/evaporation should produce narrower spectra. At sufficiently

large r, the 2nd term with v(r) in (4.9) dominates and the slope is

)2()(

2 +−=

v

ss BbS

rvαβ . (4.13)

Since fs should decrease at large r, the slope βs2 should be positive. The terms αs and S should

have opposite signs, i.e., αs < 0 and LWC (IWC) increases downward in the growth layer (S > 0,

αs < 0), and decreases downward in the evaporation layer (S < 0, αs > 0), which is physically

justified for this limit v(r) >> w. The argument in the exponent is 2222 ~)(~ +vB

s rrvrrβ , i.e., the

tails of the spectra with v(r) ~ r2, ~r and ~r1/2 (e.g., Rogers 1979; KC05), decrease as exp(-r4),

exp(-r3), and exp(-r2.5), respectively. That is, a larger value of v(r) is associated with a greater

slope and shorter tail, which is consistent with increased precipitation from the tail.

19

In the limit when v(r) is small relative to the other terms (e.g., v(r) << |w|), the exponent

βs2 is determined by the 1st term in parentheses (4.9); neglecting for simplicity w′, k′, αs′, we

obtain

)(2

1 22 kw

bS scolss ασαβ −+= . (4.14)

Again, the signs of S and of expression in parentheses should coincide to obtain βs2 > 0. In

particular, if S > 0, w > 0, and the term αsw is greater than σcol and αs2k in (4.14), then αs > 0, i.e.,

qls increases with height, which is justified for this limit w >> v(r). In this case, the slopes become

steeper (spectrum narrows) when w increases (as in regular condensation), and when σcol

increases (faster absorption by the large fraction), and βs2 decreases when k increases (turbulence

causes broadening of the spectra). Since βs2 does not depend on r for this case, the tail of the

spectrum decreases as exp(-r2).

The merged solution (4.11) differs from those found previously in the works cited in

Introduction in the following ways. At small r, the solution is a power law with variable rather

than fixed index p, allowing variable spectral dispersions. Also, the slope of the tail is not

constant but varies (generally increases) with r; accounting for the greater depletion of large

particles with greater sedimentation rate and includes depletion of the small fraction due to

accumulation by the large fraction. The tail (4.11) at large r behaves as exp(-rλ) with λ varying

from 2 to 4 for various situations, in broad agreement with the range of values determined in

previous analytical solutions.

5. Physical interpretation of the parameters

The general equation (3.4) shows that the index p ~ S, and hence p can be positive or

negative. These two cases are considered below.

a) p > 0.

20

If the major source of supersaturation is uplift and radiative cooling, then p can be

expressed similar to KC99a,b via the “effective” vertical velocity wef, using (2.19) - (2.21),

kGcwr

kGc

wcp

nn

ef

nn

efcon == 2 . (5.1)

The term “effective” here means subgrid velocities with addition of the “radiative-effective”

velocities wrad introduced in KC99a,b that allow direct comparison of the dynamical and radiative

cooling rates

radef www +≈ ' , radarad tTw )/)(/1( ∂∂−= γ , (5.2)

where (∂T/∂t)rad is the radiative cooling rate caused by the total radiative flux divergence.

Morrison et al. (2005a, b) and Morrison and Pinto (2005) found that (5.1) and (5.2) can be

successfully used to evaluate p-indices of the gamma distributions in a bulk cloud model. They

specified the subgrid velocities w′ via the turbulent coefficient k and mixing length lmix:

mixlkw /'≈ . (5.3)

This formulation for w′ is similar to that in Ghan et al. (1997). Using (2.17) - (2.19) and assuming

cnn = 1, the index p from (5.1) can be written also in the following forms

a

w

sdp

ef rNc

Lk

wp

ρπρ

γγ

34)( −

= zq

qk

wdzdq

qk

w

adls

lsef

adls

lsef

)/( ,,

≈≈ . (5.4)

If wef > 0, then p is positive. The expressions (5.4) for this case show that p depends on 4

factors: w, k, adiabatic liquid (ice) ratio X = qL/ql,ad, and height above cloud base z. This means

that p increases (spectra narrow) when positive wef increases, which is similar to the effect of

regular condensation, and p decreases (spectra broaden) when k increases, which describes the

effect of turbulence in stochastic condensation. The additional factor in (5.4) is the adiabatic

liquid (ice) ratio, X. In liquid clouds, X usually decreases with height (e.g., PK97), causing

broadening of the spectra with height. The height-factor z causes a counter-effect, causing

increase of p and spectral narrowing. Which effect will dominate depends on the product Xz. If X

21

decreases with height faster than z, then the spectra will broaden with height as observed in a

convective cloud by Warner (1969) and if X decreases with height more slowly than z, then the

spectra will narrow with height, resembling the effect of regular condensation, as observed also in

a convective cloud by Austin et al. (1985). If the product Xz is close to but slightly lower than the

adiabatic value over some height interval, such as in convective adiabatic cores due to lateral

mixing (i. e., Xz ≈ const), then the spectra can be narrow, but still wider than would be in

adiabatic ascent, similar to the situation observed by Brenguier and Chaumat (2001).

The values of p calculated in KC99b and in Morrison et al. (2005a) with (5.4) for liquid

arctic stratus cloud described in Curry (1986) were in the range 5-10 with corresponding relative

spectral dispersions from (1.3) σr = 0.3 - 0.4, close to the values observed in Curry (1986). The

values of p (σr) calculated in KC99b for a convective cloud described in Austin et al. (1985)

varied from 13-25 (σr = 0.2 - 0.3) in the middle of the cloud to 22 -35 (σr = 0.17 - 0.21) near

cloud top. Thus, the spectra are narrowing with height in this case, in contrast to the convective

cloud observed by Warner (1969), where spectra broadened with increasing height. Note that

these observed values of σr calculated with this formulation for p are substantially lower than σr =

const = 0.71 with p = 1 and σr = 0.58 with p = 2 predicted by the most cited theories of stochastic

condensation.

b) p < 0.

Eq. (3.4) predicts that the index p is negative in the cloud layers with subsaturation, S < 0.

Subsaturation may occur in a cloud due to downdrafts, advection of drier or warmer air,

entrainment of dry ambient air, and radiative heating. Under these conditions, the smallest

particles should have been evaporated to some boundary values r* and the left branch is described

by the inverse power law ps rrrf )/(~)( *

||* )/(~ prr − , which coincides with (1.4). Such inverse

power laws have been found by fitting experimental data from crystalline (cirrus and frontal)

clouds in HP84, Platt (1997), and Ryan (2000). Here, this inverse power law was obtained as a

22

solution to the kinetic equation and its index p is expressed in (3.4) via physical cloud parameters.

An estimation for cirrus clouds shows that at T ~ -40ºC with k = 5 - 10 m2 s-1 and G ~ 1- 2×10-7,

assuming cnn = 1, the parameters are kG2 ~ 0.5-1×10-9 cm2 s-1, b ~ 4×10-8 cm2 s-1. Then at

subsaturation S = - 0.1 (- 10 %) we obtain p = bS/kG2 ~ -2 - 4. The indices estimated here are

quite comparable to those in HP84, Platt (1997), and Ryan (2000) that give p ~ -2 to -5.

Note that the subsaturated layers may constitute significant portions of crystalline clouds,

particularly as falling particles enter the subsaturated regions below the cloud. Hall and

Pruppacher (1976) showed that falling crystals can survive at subsaturation over 2-6 km, which

was confirmed by later modeling studies (e.g., Jensen et al. 1994; Khvorostyanov and Sassen

1998, 2002, hereafter KS98, KS02; Khvorostyanov, Curry et al. 2001). The life cycle of cirrus

clouds usually begins with ice nucleation on haze particles that requires some threshold

supersaturation. After this threshold has been reached, the rate of vapor absorption by the crystals

may exceed the rate of supersaturation generation, so that the mean supersaturation decreases, the

thickness of cloud evaporation layer increases and can exceed the thickness of the growth layer.

This process, governed by the crystal supersaturation relaxation time, is slow and may last up to

several hours. This may cause significant frequency of ice subsaturated (evaporation) layers in

crystalline clouds and explain observations of the inverse power law spectra of HP84 type, based

on the results of this section.

6. Calculation of size spectra for a crystalline cloud

Examples of calculations of the size spectra for liquid cloud based on stochastic

condensation theory were given in KC99b, and here we present calculations of the spectra and

their parameters for a crystalline cloud. Calculations are performed using 4 different input

profiles. The baseline profiles approximate a 2 -layer cloud of cirrus and nimbostratus observed

on July 8 1998 in SHEBA-FIRE field campaign in the Arctic and simulated with bin

23

microphysics model in Khvorostyanov, Curry et al. (2001, hereafter KCetal01). The cloud shown

in Fig. 1 is pure crystalline, the small fraction exists in the region between 4 and 10 km with

maximum IWC qls = 20 mg m-3 at 7 km and maximum crystal concentration Ni ~ 90 L-1 at 9 km;

and the mean crystal radius r increases from 30 µm near cloud top to about 90 µm at the bottom.

The ice supersaturation is assumed constant at S = -15 % below 7.2 km, increasing from 0 at 7.2

km to maximum of 12 % at 9 km and then decreasing to - 80 % (Fig. 1c). This design of S is

consistent with observations (e.g. Jensen et al. 2001) and detailed simulations using bin

microphysics and supersaturation equation in KCetal01, KS98 and KS02, which showed that the

maximum IWC in cirrus can be located near S = 0, since IWC increases downward in all the

layer with S > 0. Maxima in S and Ni may coincide, reflecting the mechanism of ice nucleation

governed by supersaturation. The values of αs calculated from (2.13) are positive below the level

of maximum IWC and are negative above, with extremes of about ± 2 km-1 (Fig. 1d). We used the

(constant) values of w = 3 cm s-1 and k = 5 m2 s-1. These profiles are referred hereafter as the

baseline (case 1) and denoted in the legends as “Base”. To test the sensitivity of the results to the

input profiles, the same calculations were performed with the following changes: doubled IWC

and r (case 2); same IWC as in the baseline case but doubled r (case 3); and same conditions as

the baseline case but with doubled k (case 4).

Profiles of the parameter G calculated with (2.17) and these input data are shown in Fig.

2a. The values of G reach a maxima ~10-5 near the cloud base, reach a minimum of 1-2× 10-7 at a

height of 9 km (coincident with maxima in Ni ) and again increase upward. In case 3 (doubled r ),

G is twice as large since G ~ r ; note the r dependence of G results in much smaller values of G

for liquid clouds (KC99b) than for ice clouds. Both case 2 and 4 show the same values of G as in

the baseline case; in case 2, the increase in IWC is compensated in (2.17) by increase in r and in

case 4 G does not depend on k.

24

Corresponding profiles of the index p calculated from (3.4) with cnn = 1 are shown in Fig.

2b. In the baseline case, the values of p are negative, from 0 to -9, in the cloud evaporation layer

with S < 0 below 7 km. Values of p are positive, ranging from 0 to 13, above in the cloud growth

layer with S > 0. These variations in p are caused by simultaneous variations in S and G. The

values of p decrease by a factor of 2 with doubled k (case 4, asterisks) since p ~ k-1; this causes an

increase in spectral dispersions σr according to (1.3) and reflects the effect of stochastic

condensation as increasing turbulence broadens the spectra. The values of p decrease by a factor

of 4 with doubled r (crosses) since p ~ G-2, which is also an effect of stochastic condensation. As

discussed in KC99a,b, the diffusion coefficient in radii-space is given by, kr ~ G2k; thus,

according to (2.17), kr ~ 2r for fixed qls, this feature resembles Taylor’s diffusion with the

diffusion coefficient k ~ l2, with l being mixing length. In stochastic condensation, r is a

characteristic length and plays the role of mixing length in radii space.

Shown in Fig. 3 are the slopes βs of calculated from (4.9) in the upper layer with ice

supersaturation at the heights 7.5 - 8.7 km and in the lower layer at heights 4.8 - 6.0 km with S <

0. The common feature in both layers is substantial increase of the slopes with increasing radius,

which arises from the dominance of the second term with v(r) in (4.9) that tends to the limit

(4.13). In the upper layer, the slopes decrease downward causing spectral broadening, which

agrees with observations. This is caused by dominance of the terms with αs, and decrease of |αs|

downward, toward the level with S = 0. In the lower layer, the values of βs2 are much smaller due

to smaller αs (see Fig. 1d) and increase downward. The last feature would cause an incorrect

narrowing of the spectra downward, but it is overwhelmed by the effect of the negative indices p

in this layer as described below.

Fig. 4 depicts size spectra fs(r) calculated with input profiles from the baseline case. The

left panels (a, c) show the tails of the spectra calculated from (4.8), and the right panels (b, d)

show the composite spectra (4.11) with multiplication by rpΦ(r) and Φ(r) from (4.12). One can

25

see that in the upper ice supersaturated layer with p > 0, the spectra resemble typical gamma

distributions. Addition of the factor rpΦ(r) allows extension of the spectra to the small radii

region but does not cause substantial change of the spectra.

The situation is different in the lower ice subsaturated layer with p < 0. Without the factor

rpΦ(r), the small size region up to 100 µm in double log coordinates represents parallel straight

lines with positive slopes caused by the factor r before exponent in (4.8) (recall, the solution (4.8)

is typical for some previous theories of stochastic condensation with p = 1). With multiplication

by rpΦ(r) in (4.11), the spectra become inverse power laws up to r ~ 40 - 60 µm, the slopes

(indices p) increase with height (i.e., with decreasing temperature); and the spectra show

bimodality with secondary maxima at r = 100 - 150 µm (Fig. 4d). Such bimodal behavior is often

observed in crystalline clouds and parameterized with superposition of the two inverse power

laws and exponents that correspond to the two size fractions (e.g., HP84, Mitchell et al. 1996;

Platt 1997, Ryan 2000). The solution (4.11) shows that this bimodality may occur within the

small-size fraction: the left branch (inverse power law) is a result of stochastic and regular

condensation, while the tail is dominated by the interaction of regular condensation, turbulent and

convective transport, and sedimentation.

7. Conclusions

The stochastic kinetic equation for the drops and ice crystal size spectra f(r) derived in

KC99a is extended here to include the accretion/aggregation processes. Under a number of

assumptions typical for the cloud bulk models, analytical solutions to the kinetic equation are

obtained here for the small-size fraction of the cloud droplet and crystal size spectra. The

solutions obtained for the small-size fraction have the form

)())(exp()( rrrrcrf pNs Φ−= λβ . (7.1)

i.e., are close to generalized gamma distributions.

26

A general yet simple analytical expression is found for the index p, which is expressed

via supersaturation, cloud integral parameters (liquid or ice water content, mean radius or number

density), and a few fundamental constants. Depending on supersaturation, the index p can be

positive, yielding gamma distributions, or negative, yielding inverse power laws. The positive

values of p can vary in the range 0 - 12 in stratiform clouds, and can reach 30-40 for convective

clouds. The range of negative p values is 0 to - 12, close to observations in liquid and crystalline

clouds. The values and profiles of p provide a physical explanation of the observed spatio-

temporal variations of the spectral dispersions observed in various cloud types. These values of p

correspond to dispersions that are smaller than in many previous theories, providing better

agreement with observations than previous theoretical values that yielded spectra broader than

observed. In evaporating layers with negative p, the expression (7.1) yields a bimodal spectra

within small-size fraction alone, and the spectra represent a superposition of the Marshall-Palmer

exponent and the Heymsfield-Platt inverse power law.

This simple expression for the index p can be easily incorporated in bulk microphysics

parameterizations instead of prescribed values for p for cloud drop and ice crystal size spectra in

cloud and climate models that include supersaturation fluctuations (e.g., Morrison et al., 2005a,b;

Morrison and Pinto 2005) or parameterize them via subgrid velocities.

The two analytical solutions obtained here differ by the account or neglect of the

condensation/deposition in the tails of the spectra, which is determined by the magnitude of sub-

or supersaturation. The slopes β(r) of the tails are simply expressed via dynamical quantities

(vertical velocity, turbulent coefficient and their derivatives), and the rate of collection of the

small particles by the larger precipitating particles. The slopes of the solution neglecting

condensation/deposition of the larger particles do not depend on radius for negligible fall

velocities and on supersaturation, and the value of λ = 1 in (7.1). The slopes of the solution

including condensation/deposition of the larger particles are inversely proportional to the

27

supersaturation and proportional at sufficiently large radii to the terminal velocity. The slopes

β(r) of the second solution depend on radius as β(r) = cβrκ with κ > 1, and the tails of the spectra

have the form exp(-cβrλ) with λ varying from 2 to 4 for various radii and regimes of fall speeds.

The range of λ in both solutions (1 - 4) encompasses the range of fixed values of λ

obtained in the other theories of size spectra cited in the Introduction (2 - 3), but in contrast to

these previous theories, the values of λ derived here are not fixed but depend on the physical

situation.

These solutions provide physical explanations for observed dependencies of the spectra

on the temperature, turbulence, liquid water or ice water content, and other cloud properties. The

results are illustrated with an example of calculations for crystalline clouds. These analytical

solutions can be used for parameterization of the size spectra and related quantities (e.g., optical

properties, radar reflectivities) in bulk cloud and climate models and in remote sensing

techniques. They can be also used for testing the accuracy of the numerical cloud models, where

the numerical diffusivity can be significant. Note that despite the existence of such kinetic

equations of stochastic condensation for more than four decades and the recent refinements to the

stochastic condensation theory (e.g., Liu 1995; Korolev 1995; Liu and Hallett 1998; KC99a,b;

Vaillancourt et al., 2002; Shaw 2003; Paoli and Shariff 2003, 2004; Tisler et al. 2005; McGraw

and Liu 2006; Jeffery et al. 2007; this work), all currently employed spectral bin models account

for only advection in radii space and neglect the diffusion by radius and cross-derivative terms;

the error of this approximations is unknown and can be significant.

Analytical solutions have been presented here for liquid only and ice only size spectra.

Treatment of mixed-phase clouds would require simultaneous consideration of the kinetic

equations for the drop and crystal spectra with account for their interactions. The analytical

solutions for the large size spectra (precipitating particles) are considered in Part 2 of this work

(KC07).

28

Acknowledgments. This research has been supported by the DOE Atmospheric Radiation

Measurement Program and NASA Modeling and Parameterization Program. Jody Norman is

thanked for help in preparing the manuscript.

29

Appendix A

Derivation and solution of the Kummer equation

In KC99b, the details of the derivation and solution of the Kummer equation for the

interpolation function Φ were omitted. Here, we present them because of potential use of this

technique for the bulk cloud models. It is convenient to rewrite composite spectrum (3.10) with

auxiliary function η(r)

)()()( rrrf s Φ=η , (A.1)

)exp()( 2,1 rrcr sp βη −= . (A.2)

First, we assume that v(r) = const, then µs = const, and βs = const, a correction for r-dependence

of the terminal velocities can be accounted for by introducing real v(r) into the final solutions.

The derivatives fs′ and fs′′ are expressed as

')/(' Φ+Φ−= ηηβrpf s , (A.3)

''')/(2)/()/('' 22 Φ+Φ−+Φ−Φ−= ηηβηηβ rprprpf s . (A.4)

Substitution of (A.3), (A.4) into (2.16) and deleting η(r) in the resulting equation yields

Φ+Φ

−+Φ−Φ

− '''22

222 ββ

rp

rkpG

rpkG

Φ+Φ

−

−+ '2 βα

rp

rbSkG s

0)1( 222 =

+−

rbSk ss µα . (A.5)

Collecting now the terms with various derivatives of Φ, we obtain an equation

0322

2

1 =Φ+Φ

+Φ A

drdA

drdA , (A.6)

where

kGA 21 = , (A.7)

30

−+−=

rbSkGkG

rkpGA ss αβ 222 2

2

2 , (A.8)

222

2

22

32

ss kG

rkpG

rkpGA β

β+−=

sss kG

rbSp

rkpG

rkpG

βαα 22

22

2−−+−

222 )1(

rbSk

rbS

sss +−++ µα

β . (A.9)

These coefficients are significantly simplified after substituting the expression bS = G2kp

that follows from (3.4) (with cnn =1) and (3.9) for βs. First, for βs1 = (αs/G)(1-µ s), we have

23

22s

sss kr

Gkpr

GkpA αµαα

++−=

rkpGkk s

ssssα

µαµα22 222 ++−

rpGkkk s

sssα

µαα ++− 22 22

222sss

ss kkr

pGkµαα

µα−+− . (A.10)

This equation contains 12 terms, but one can see that the following 9 terms are mutually

cancelled: 1st and 6th; 4th and 8th; 3rd, 7th and 11th; 5th and 12th. The rest 3 terms yields

)1(3 ss

rpGkA µ

α+= (A.11)

The coefficient A2 is also simplified after substitution of bS via p and βs1:

rkpGkG

GkG

rkpGA ss

s2

22

2 2)1(22 −+−−= αµα

+=

GrpkG ssµα22 . (A.12)

31

Substituting these reduced A2, A3 into (A.6), dividing by G2k and multiplying by r we obtain the

equation for βs,1

0)1(2=++Φ′

++Φ ′′ s

sssG

prG

pr µαµα . (A.13)

A solution to this equation is the confluent hypergeometric (Kummer) function (Landau and

Lifshitz 1958, hereafter LL58; Gradshteyn and Ryzhik 1994, hereafter GR94)

−+=Φ r

GppFcr ss

ss

µαµ

µ2;),1(

2)( 1 . (A.14)

Then the full solution for fs is

−−= )1(exp)( 11, s

sps G

rcrf µα ,

−+× r

GppF ss

ss

µαµ

µ2;),1(

2, (A.15)

For βs2 = (αs/G)(1+µ s), substituting it and bS = G2kp into (A.9) for A3, we obtain

23

22s

sss kr

Gkpr

GkpA αµαα

+−−=

rkpGkk s

ssssα

µαµα22 222 +++

rpGkkk s

sssα

µαα +−− 22 22

222sss

ss kkr

pGkµαα

µα−++ . (A.16)

Here also, the following terms are mutually cancelled: 1st and 6th; 3, 7th and 11th; 4th and 8th;

5th and 12th. The rest 3 terms yield

)1(3 ss

rpGkA µ

α−= . (A.17)

The coefficient A2 in (A.8) is simplified after substitution of bS via p and βs2:

32

rkpGkG

GkG

rkpGA ss

s2

22

2 2)1(22 −++−= αµα

+=

GrpkG ssµα22 . (A.18)

Again substituting reduced A2, A3 into (A.6), dividing by G2k and multiplying by r we obtain the

equation for βs,2

0)1(2=−+Φ′

−+Φ ′′ s

sssG

prG

pr µαµα . (A.19)

A solution to this equation is the confluent hypergeometric function (GR94)

+−=Φ r

GppFcr ss

ss

µαµ

µ2;),1(

2)( 2 . (A.20)

The full solution for fs is

+−= )1(exp)( 22, s

sps G

rcrf µα ,

+−× r

GppF ss

ss

µαµ

µ2;),1(

2, (A.21)

The confluent hypergeometric functions F(a,b;x) have the following properties. The

function F(a,b;x) can be defined as a power series

...!2)1(

)1(!1

1),,(2

+++

++=x

bbaax

baxbaF (A.22)

This series is convenient for coding, rather rapidly converges and usually does not require a large

number of terms. A few useful limits follow from this definition.

1);,0( =xbF , ( ) 1,,lim0

=→

xbaFx

, xexaaF =),,( (A.23)

For analysis of asymptotic behavior of F(a,b;x) at large x, another representation of this function

is used as a contour integral over a contour C in the complex x-plain (LL58)

33

∫ −−−Γ

=C

baat dttxtei

bxbaF )(2

)();,(π

, (A.24)

where Γ(b) is the Euler gamma function. The subintegral function in (A.24) has 2 singular points,

at x = t and x = 0. Deforming contour C in such a way that it transforms into 2 contours each

going in the counter clock direction around these 2 singular points, dividing subintegral function

by (-x)-a, expanding it by the powers t/x and integrating all terms, the following expression is

obtained

);1,()()(

)();,( xbaaxab

bxbaF a −+−Υ−−Γ

Γ= −

);1,()()( xaabxe

ab bax −−−Υ

ΓΓ

+ − , (A.25)

where the function Y(a,b;x) is a power series by the inverse powers of x

...!2

)1))1(!1

1);,( 2 +++

++=Υx

bbaax

abxba (A.26)

It is easy to see that Y(a,b;x) → 1 when x → ∞. Using this feature with (A.25), we obtain the

following limit

( ))()(,,lim bax

xxe

abxbaF −

∞→ ΓΓ

≈ . (A.27)

This limit is very convenient for analysis of the asymptotic behavior of the size spectra at large

radii.

The spectral moments for the distribution functions (e.g., number density, mean radius,

liquid or ice water content) can be evaluated using the relations (LL58; GR94)

∫∞

−=0

)( ),,( dxxbaFexM x βλνν

++Γ= +−

λβ

νλν ν ,,1,)1( )1( baF , (A.28)

34

where F(α,β,γ;x) is the Gauss hypergeometric function. Using this equation, we get the following

moments for the 1st solution (A.15)

∫∞

=0

)( )( drrfrM snn

( ))1(

1 1)1(++−

−++Γ=

np

sGnpc µ

α

−

−+++×s

ss pnppF

µµ

µµ 1

2;,1),1(2

, (A.29)

and for the 2nd solution (A.21)

∫∞

=0

)( )( drrfrM snn

)1(

2 )1()1(++−

+++Γ=

np

sGnpc µ

α

+

++−×s

ss pnppF

µµ

µµ 1

2,,1),1(2

. (A.30)

In particular, for n = 0, M(0) = N and we obtain normalization constants c1, c2. For solution

(A.15), we obtain:

( ) ( )( )

( )11

1 12,,1,1

211

−+−

−

−++

−+Γ=

s

ss

p

s pppFG

pNcµ

µµ

µµα (A.31)

For (A.21) we obtain

( ) ( )( )

( )11

2 12,,1,1

211

−+−

+

+−

++Γ=

s

ss

p

s pppFG

pNcµ

µµ

µµα (A.32)

These relations can be used for evaluation of the LWC (IWC), number densities, mean radii,

extinctions coefficients and other moments of the solutions given in section 3.

35

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42

Figure captions

Fig. 1. Vertical profiles of the input parameters for the 4 different cases indicated in the legend

and described in the text (a) ice water content, mg m-3; (b) crystal concentration, L-1; (c) ice

supersaturation, %; (d) the parameter αs, km-1.

Fig. 2. Profiles of the parameter G (a) and (b) the index p of size spectra.

Fig. 3. Slopes βs determined from (4.9) at various altitudes in the layers with: (a) ice

supersaturation and (b) subsaturation. The heights in km are indicated in the legends.

Fig. 4. Size spectra calculated for the baseline case for the heights indicated in the legends. a)

supersaturated layer using (4.10) for the tail; b) supersaturated layer using (4.11) for the

composite spectra with multiplication by rpΦ(r); c) subsaturated layer using (4.10) for the tail; d)

b) supersaturated layer using (4.11) for the composite spectra with multiplication by rpΦ(r).

0 5 10 15 20 25 30 35 40 45IWC (mg m-3)

3

4

5

6

7

8

9

10

11

12H

eigh

t(km

)(a)

Base

2 IWC

2R

2K

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20Ice supersaturation (%)

3

4

5

6

7

8

9

10

11

12

Hei

ght(

km)

(c)

0 25 50 75 100Crystal concentration (L-1)

3

4

5

6

7

8

9

10

11

12

Hei

ght(

km)

(b)

Base

2IWC

2R

2K

-3 -2 -1 0 1 2αs (km-1)

3

4

5

6

7

8

9

10

11

12

Hei

ght(

km)

(d)

Fig. 1. Vertical profiles of the input parameters for the 4 different cases indicated in the legend and described in the text (a) ice water content, mg m-3; (b) crystal concentration, L-1; (c) ice supersaturation, %; (d) the parameter αs, km-1.

1E-7 1E-6 1E-5 1E-4Parameter G

3

4

5

6

7

8

9

10

11

12

Hei

ght(

km)

(a)

Base

2IWC

2R

2K

-12 -9 -6 -3 0 3 6 9 12 15Index p

3

4

5

6

7

8

9

10

11

12

Hei

ght(

km)

(b)

Base

2IWC

2R

2K

Fig. 2. Profiles of the parameter G (a) and (b) the index p of size spectra.

0 100 200 300 400 500 600Crystal radius (µm)

0E+0

2E+4

4E+4

6E+4

8E+4

1E+5S

lope

β s2

(cm

-2)

(a)S > 0

7.5 km

7.8 km

8.1 km

8.4 km

8.7 km

0 100 200 300 400 500 600Crystal radius (µm)

0E+0

4E+3

8E+3

1E+4

2E+4

2E+4

Slo

peβ s

2(c

m-2

)

(b)S < 0

4.8 km

5.1 km

5.4 km

5.7 km

6.0 km

Fig. 3. Slopes βs determined from (4.9) at various altitudes in the layers with: (a) ice supersaturation and (b) subsaturation. The heights in km are indicated in the legends.

0 100 200 300 400Crystal radius (µm)

1E-3

1E-2

1E-1

1E+0

1E+1

Cry

stal

size

spec

tra(L

-1µ m

-1)

(a) 7.5 km

7.8 km

8.1 km

8.4 km

8.7 km

1 10 100Crystal radius (µm)

1E-4

1E-3

1E-2

1E-1

1E+0

1E+1

Cry

stal

size

spec

tra(L

-1µm

-1)

(c)

4.8 km

5.1 km

5.4 km

5.7 km

6.0 km

0 100 200 300 400Crystal radius (µm)

1E-3

1E-2

1E-1

1E+0

1E+1

Cry

stal

size

spec

tra(L

-1µ m

-1)

(b) 7.5 km

7.8 km

8.1 km

8.4 km

8.7 km

1 10 100Crystal radius (µm)

1E-4

1E-3

1E-2

1E-1

1E+0

1E+1

1E+2

Cry

stal

size

spec

tra(L

-1µm

-1)

(d)

4.8 km

5.1 km

5.4 km

5.7 km

6.0 km

Fig. 4. Size spectra calculated for the baseline case for the heights indicated in the legends. a) supersaturated layer using (4.10) for the tail; b) supersaturated layer using (4.11) for the composite spectra with multiplication by rpΦ(r); c) subsaturated layer using (4.10) for the tail; d) b) supersaturated layer using (4.11) for the composite spectra with multiplication by rpΦ(r).